Räuberzahlen

Räuberzahlen, also known as parasitic numbers, are a peculiar concept in number theory. A Räuberzahl is a positive integer that, when multiplied by another integer, produces a product containing the same digits as the original number, just in a different order. For example, 102564 multiplied by 4 equals 410256. Both numbers use the same set of digits {0, 1, 2, 4, 5, 6}.

The smallest known Räuberzahl is 102564. Other examples include 1025640, 10256400, and so on, by appending zeros. These are often called trivial Räuberzahlen. The non-trivial Räuberzahlen are more elusive. The next smallest non-trivial Räuberzahl after 102564 is 1008403245.

The existence of Räuberzahlen is related to the properties of permutations of digits and the concept of digital roots. If a number and its permutation have the same digital root, and the multiplier is such that the digits are merely rearranged, then it qualifies as a Räuberzahl.

Research into Räuberzahlen continues, with mathematicians exploring their properties and searching for larger examples. The concept provides an interesting intersection of arithmetic and combinatorics.